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Simplification of division of integral of standard normal distribution

Mathematics Asked by Ga13 on November 29, 2021

I’m integrating two standard normal distributions and, for numerical problems, I would like to simplify the expresion below, since the only difference between the numerator and denominator is the upper limit, where $theta$, $beta in mathbb{R}$.

$$
cfrac{int _{-infty }^{theta}!{frac {{{rm e}^{-{x}^{2}/2,}}}{ sqrt{2,pi }}}{dx}}{ int _{-infty }^{beta}!{frac {{{rm e}^{-{x}^{2}/2,}}}{ sqrt{2,pi }}}{dx}}.
$$

Any hint on how can I do this simplification?

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